Thermalization of Levy flights: Path-wise picture in 2D

TL;DR

This paper investigates the thermalization process of 2D Levy flights driven by symmetric Levy noise, using a modified Gillespie algorithm to approximate solutions that converge to a specified Boltzmann-type equilibrium distribution.

Contribution

It introduces a path-wise simulation method for Levy flights responding to potential landscapes, enabling the study of their convergence to equilibrium distributions beyond traditional Langevin models.

Findings

Successful adaptation of Gillespie's algorithm for Levy noise in 2D.

Demonstrated convergence of sample trajectories to target Boltzmann distributions.

Analyzed response of 2D Cauchy noise to periodic potential landscapes.

Abstract

We analyze two-dimensional (2D) random systems driven by a symmetric L\'{e}vy stable noise which, under the sole influence of external (force) potentials $\Phi (x) $, asymptotically set down at Boltzmann-type thermal equilibria. Such behavior is excluded within standard ramifications of the Langevin approach to L\'{e}vy flights. In the present paper we address the response of L\'{e}vy noise not to an external conservative force field, but directly to its potential $\Phi (x)$. We prescribe a priori the target pdf $\rho_*$ in the Boltzmann form $\sim \exp[- \Phi (x)]$ and next select the L\'evy noise of interest. Given suitable initial data, this allows to infer a reliable path-wise approximation to a true (albeit analytically beyond the reach) solution of the pertinent master equation, with the property $\rho (x,t)\rightarrow \rho_*(x)$ as time $t$ goes to infinity. We create a suitably modified version of the time honored Gillespie's algorithm, originally invented in the chemical kinetics context. A statistical analysis of generated sample trajectories allows us to infer a surrogate pdf dynamics which consistently sets down at a pre-defined target pdf. We pay special attention to the response of the 2D Cauchy noise to an exemplary locally periodic "potential landscape" $\Phi (x), x\in R^2$.